Large Scale Conditional Covariance Matrix Modeling, Estimation and Testing

A new representation of the diagonal Vech model is given using the Hadamard product. Sufficient conditions on parameter matrices are provided to ensure the positive definiteness of covariance matrices from the new representation. Based on this, some new and simple models are discussed. A set of diagnostic tests for multivariate ARCH models is proposed. The tests are able to detect various model misspecifications by examing the orthogonality of the squared normalized residuals. A small Monte-Carlo study is carried out to check the small sample performance of the test. An empirical example is also given as guidance for model estimation and selection in the multivariate framework. For the specific data set considered, it is found that the simple one and two parameter models and the constant conditional correlation model perform fairly well.